Properties

Label 45T32
Order \(270\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $He_3:D_5$

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Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $32$
Group :  $He_3:D_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,29)(2,30)(3,28)(4,26,5,25,6,27)(7,23,9,22,8,24)(10,19)(11,21)(12,20)(13,18,15,17,14,16)(31,45,33,44,32,43)(34,40,35,42,36,41)(37,39), (1,9,3,8,2,7)(4,5)(10,44,12,43,11,45)(13,40)(14,42)(15,41)(16,38,18,39,17,37)(19,34,21,35,20,36)(22,32)(23,31)(24,33)(25,29,27,30,26,28)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
10:  $D_{5}$
18:  $S_3\times C_3$
54:  $C_3^2 : C_6$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $S_3$

Degree 5: $D_{5}$

Degree 9: $C_3^2 : C_6$

Degree 15: $D_{15}$

Low degree siblings

There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $3$ $( 4, 5, 6)( 7, 9, 8)(13,15,14)(16,18,17)(22,24,23)(25,27,26)(31,33,32) (34,35,36)(40,42,41)(43,45,44)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $3$ $( 4, 6, 5)( 7, 8, 9)(13,14,15)(16,17,18)(22,23,24)(25,26,27)(31,32,33) (34,36,35)(40,41,42)(43,44,45)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $45$ $2$ $( 2, 3)( 4,43)( 5,45)( 6,44)( 7,42)( 8,40)( 9,41)(10,37)(11,38)(12,39)(13,36) (14,35)(15,34)(16,33)(17,31)(18,32)(19,30)(20,28)(21,29)(22,25)(23,26)(24,27)$
$ 6, 6, 6, 6, 6, 2, 2, 2, 2, 2, 2, 2, 1 $ $45$ $6$ $( 2, 3)( 4,44, 5,43, 6,45)( 7,40, 9,42, 8,41)(10,37)(11,38)(12,39) (13,35,15,36,14,34)(16,31,18,33,17,32)(19,30)(20,28)(21,29)(22,26,24,25,23,27)$
$ 6, 6, 6, 6, 6, 2, 2, 2, 2, 2, 2, 2, 1 $ $45$ $6$ $( 2, 3)( 4,45, 6,43, 5,44)( 7,41, 8,42, 9,40)(10,37)(11,38)(12,39) (13,34,14,36,15,35)(16,32,17,33,18,31)(19,30)(20,28)(21,29)(22,27,23,25,24,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,14,15)(16,18,17)(19,20,21) (22,23,24)(25,27,26)(28,30,29)(31,32,33)(34,35,36)(37,39,38)(40,41,42) (43,45,44)$
$ 15, 15, 15 $ $6$ $15$ $( 1, 4, 7,11,15,16,21,22,25,29,33,34,38,42,43)( 2, 6, 9,12,13,18,19,23,27,28, 31,35,37,40,45)( 3, 5, 8,10,14,17,20,24,26,30,32,36,39,41,44)$
$ 15, 15, 15 $ $6$ $15$ $( 1, 4, 8,11,15,17,21,22,26,29,33,36,38,42,44)( 2, 6, 7,12,13,16,19,23,25,28, 31,34,37,40,43)( 3, 5, 9,10,14,18,20,24,27,30,32,35,39,41,45)$
$ 15, 15, 15 $ $6$ $15$ $( 1, 4, 9,11,15,18,21,22,27,29,33,35,38,42,45)( 2, 6, 8,12,13,17,19,23,26,28, 31,36,37,40,44)( 3, 5, 7,10,14,16,20,24,25,30,32,34,39,41,43)$
$ 15, 15, 15 $ $6$ $15$ $( 1, 7,13,21,25,31,38,43, 6,11,16,23,29,34,40)( 2, 9,14,19,27,32,37,45, 5,12, 18,24,28,35,41)( 3, 8,15,20,26,33,39,44, 4,10,17,22,30,36,42)$
$ 15, 15, 15 $ $6$ $15$ $( 1, 7,15,21,25,33,38,43, 4,11,16,22,29,34,42)( 2, 9,13,19,27,31,37,45, 6,12, 18,23,28,35,40)( 3, 8,14,20,26,32,39,44, 5,10,17,24,30,36,41)$
$ 15, 15, 15 $ $6$ $15$ $( 1, 7,14,21,25,32,38,43, 5,11,16,24,29,34,41)( 2, 9,15,19,27,33,37,45, 4,12, 18,22,28,35,42)( 3, 8,13,20,26,31,39,44, 6,10,17,23,30,36,40)$
$ 15, 15, 5, 5, 5 $ $6$ $15$ $( 1,10,19,29,39, 2,11,20,28,38, 3,12,21,30,37)( 4,13,24,33,40, 5,15,23,32,42, 6,14,22,31,41)( 7,16,25,34,43)( 8,17,26,36,44)( 9,18,27,35,45)$
$ 15, 15, 15 $ $2$ $15$ $( 1,10,19,29,39, 2,11,20,28,38, 3,12,21,30,37)( 4,14,23,33,41, 6,15,24,31,42, 5,13,22,32,40)( 7,17,27,34,44, 9,16,26,35,43, 8,18,25,36,45)$
$ 15, 15, 5, 5, 5 $ $6$ $15$ $( 1,10,19,29,39, 2,11,20,28,38, 3,12,21,30,37)( 4,15,22,33,42)( 5,14,24,32,41) ( 6,13,23,31,40)( 7,18,26,34,45, 8,16,27,36,43, 9,17,25,35,44)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,11,21,29,38)( 2,12,19,28,37)( 3,10,20,30,39)( 4,15,22,33,42) ( 5,14,24,32,41)( 6,13,23,31,40)( 7,16,25,34,43)( 8,17,26,36,44) ( 9,18,27,35,45)$
$ 15, 15, 15 $ $2$ $15$ $( 1,12,20,29,37, 3,11,19,30,38, 2,10,21,28,39)( 4,13,24,33,40, 5,15,23,32,42, 6,14,22,31,41)( 7,18,26,34,45, 8,16,27,36,43, 9,17,25,35,44)$
$ 15, 15, 15 $ $6$ $15$ $( 1,13,27,38, 6,18,29,40, 9,21,31,45,11,23,35)( 2,14,26,37, 5,17,28,41, 8,19, 32,44,12,24,36)( 3,15,25,39, 4,16,30,42, 7,20,33,43,10,22,34)$
$ 15, 15, 15 $ $6$ $15$ $( 1,13,25,38, 6,16,29,40, 7,21,31,43,11,23,34)( 2,14,27,37, 5,18,28,41, 9,19, 32,45,12,24,35)( 3,15,26,39, 4,17,30,42, 8,20,33,44,10,22,36)$
$ 15, 15, 15 $ $6$ $15$ $( 1,13,26,38, 6,17,29,40, 8,21,31,44,11,23,36)( 2,14,25,37, 5,16,28,41, 7,19, 32,43,12,24,34)( 3,15,27,39, 4,18,30,42, 9,20,33,45,10,22,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1,16,32)( 2,18,33)( 3,17,31)( 4,19,35)( 5,21,34)( 6,20,36)( 7,24,38) ( 8,23,39)( 9,22,37)(10,26,40)(11,25,41)(12,27,42)(13,30,44)(14,29,43) (15,28,45)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1,16,31)( 2,18,32)( 3,17,33)( 4,20,36)( 5,19,35)( 6,21,34)( 7,23,38) ( 8,22,39)( 9,24,37)(10,26,42)(11,25,40)(12,27,41)(13,29,43)(14,28,45) (15,30,44)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1,16,33)( 2,18,31)( 3,17,32)( 4,21,34)( 5,20,36)( 6,19,35)( 7,22,38) ( 8,24,39)( 9,23,37)(10,26,41)(11,25,42)(12,27,40)(13,28,45)(14,30,44) (15,29,43)$
$ 15, 15, 5, 5, 5 $ $6$ $15$ $( 1,19,39,11,28, 3,21,37,10,29, 2,20,38,12,30)( 4,22,42,15,33)( 5,24,41,14,32) ( 6,23,40,13,31)( 7,26,45,16,36, 9,25,44,18,34, 8,27,43,17,35)$
$ 15, 15, 15 $ $2$ $15$ $( 1,19,39,11,28, 3,21,37,10,29, 2,20,38,12,30)( 4,23,41,15,31, 5,22,40,14,33, 6,24,42,13,32)( 7,27,44,16,35, 8,25,45,17,34, 9,26,43,18,36)$
$ 15, 15, 5, 5, 5 $ $6$ $15$ $( 1,19,39,11,28, 3,21,37,10,29, 2,20,38,12,30)( 4,24,40,15,32, 6,22,41,13,33, 5,23,42,14,31)( 7,25,43,16,34)( 8,26,44,17,36)( 9,27,45,18,35)$
$ 15, 15, 15 $ $2$ $15$ $( 1,20,37,11,30, 2,21,39,12,29, 3,19,38,10,28)( 4,24,40,15,32, 6,22,41,13,33, 5,23,42,14,31)( 7,26,45,16,36, 9,25,44,18,34, 8,27,43,17,35)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,21,38,11,29)( 2,19,37,12,28)( 3,20,39,10,30)( 4,22,42,15,33) ( 5,24,41,14,32)( 6,23,40,13,31)( 7,25,43,16,34)( 8,26,44,17,36) ( 9,27,45,18,35)$
$ 15, 15, 15 $ $6$ $15$ $( 1,22,43,21,42,16,38,15,34,11,33, 7,29, 4,25)( 2,23,45,19,40,18,37,13,35,12, 31, 9,28, 6,27)( 3,24,44,20,41,17,39,14,36,10,32, 8,30, 5,26)$
$ 15, 15, 15 $ $6$ $15$ $( 1,22,44,21,42,17,38,15,36,11,33, 8,29, 4,26)( 2,23,43,19,40,16,37,13,34,12, 31, 7,28, 6,25)( 3,24,45,20,41,18,39,14,35,10,32, 9,30, 5,27)$
$ 15, 15, 15 $ $6$ $15$ $( 1,22,45,21,42,18,38,15,35,11,33, 9,29, 4,27)( 2,23,44,19,40,17,37,13,36,12, 31, 8,28, 6,26)( 3,24,43,20,41,16,39,14,34,10,32, 7,30, 5,25)$

Group invariants

Order:  $270=2 \cdot 3^{3} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [270, 14]
Character table: Data not available.